Yes there are games stores that will accept PS4 controllers. However I would highly advise against it as the games stores will rip you off considerably by paying you half of what it is worth. For example if your PS4 controller is brand new it will cost between £35-40 however the stores will buy the controller off you for say £20 and you don't get physical money they offer 'in store credit' to use in their stores. I would advise either keeping it for a friend to use if you decide to play a multiplayer game or to sell it on eBay where you will make the most amount of money.

Yes but I don"t know why you would...is it broken? The best you can trade with the ps4 controller as of 2018, (*i'm assuming at gamestop), is an Xbox One controller or "SHITTY" store credits. My opinion? Keep it around as backup.